The Annals of Applied Statistics

Bayesian sensitivity analysis for causal effects from $2\times2$ tables in the presence of unmeasured confounding with application to presidential campaign visits

Luke Keele and Kevin M. Quinn

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Presidents often campaign on behalf of candidates during elections. Do these campaign visits increase the probability that the candidate will win? While one might attempt to answer this question by adjusting for observed covariates, such an approach is plagued by serious data limitations. In this paper we pursue a different approach. Namely, we ask: what, if anything, should one infer about the causal effect of a presidential campaign visit using a simple cross-tabulation of the data? We take a Bayesian approach to this problem and show that if one is willing to use substantive information to make some (possibly weak) assumptions about the nature of the unmeasured confounding, sharp posterior estimates of causal effects are easy to calculate. Using data from the 2002 midterm elections, we find that, under a reasonable set of assumptions, a presidential campaign visit on the behalf of congressional candidates helped those candidates win elections.

Article information

Ann. Appl. Stat. Volume 11, Number 4 (2017), 1974-1997.

Received: August 2016
Revised: April 2017
First available in Project Euclid: 28 December 2017

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Causal inference Bayesian statistics partial identification sensitivity analysis


Keele, Luke; Quinn, Kevin M. Bayesian sensitivity analysis for causal effects from $2\times2$ tables in the presence of unmeasured confounding with application to presidential campaign visits. Ann. Appl. Stat. 11 (2017), no. 4, 1974--1997. doi:10.1214/17-AOAS1048.

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  • Angrist, J. D., Imbens, G. W. and Rubin, D. B. (1996). Identification of causal effects using instrumental variables. J. Amer. Statist. Assoc. 91 444–455.
  • Balke, A. and Pearl, J. (1997). Bounds on treatment effects from studies with imperfect compliance. J. Amer. Statist. Assoc. 92 1171–1176.
  • Chickering, D. M. and Pearl, J. (1997). A clinician’s tool for analyzing non-compliance. Computing Science and Statistics 29 424–431.
  • Cohen, J. E., Krassa, M. A. and Hamman, J. A. (1991). The impact of presidential campaigning on midterm U.S. senate elections. Am. Polit. Sci. Rev. 85 165–178.
  • Cole, S. R. and Hernán, M. A. (2008). Constructing inverse probability weights for marginal structural models. Am. J. Epidemiol. 168 656–664.
  • Cornfield, J., Haenszel, W., Hammond, E., Lilienfeld, A., Shimkin, M. and Wynder, E. (1959). Smoking and lung cancer: Recent evidence and a discussion of some questions. J. Natl. Cancer Inst. 22 173–203.
  • Cox, D. R. (1958). Planning of Experiments. Wiley, New York.
  • Ding, P. and Dasgupta, T. (2016). A potential tale of two-by-two tables from completely randomized experiments. J. Amer. Statist. Assoc. 111 157–168.
  • Ding, P. and VanderWeele, T. J. (2016a). Sensitivity analysis without assumptions. Epidemiology 27 368.
  • Ding, P. and VanderWeele, T. J. (2016b). Sharp sensitivity bounds for mediation under unmeasured mediator-outcome confounding. Biometrika 103 483–490.
  • Gelman, A., Carlin, J. B., Stern, H. S. and Rubin, D. B. (2004). Bayesian Data Analysis, 2nd ed. Chapman & Hall, Boca Raton, FL.
  • Gill, J. (2007). Bayesian Methods: A Social and Behavioral Science Approach, 2nd ed. Chapman & Hall/CRC, Boca Raton, FL.
  • Gill, J. and Walker, L. D. (2005). Elicited priors for Bayesian model specifications in political science research. The Journal of Politics 67 841–872.
  • Glynn, A. N. and Quinn, K. M. (2011). Why process matters for causal inference. Polit. Anal. 19 273–286.
  • Grilli, L. and Mealli, F. (2008). Nonparametric bounds on the causal effect of university studies on job opportunities using principal stratification. J. Educ. Behav. Stat. 33 111–130.
  • Gustafson, P. (2010). Bayesian inference for partially identified models. Int. J. Biostat. 6 Art. 17, 20.
  • Gustafson, P., McCandless, L. C., Levy, A. R. and Richardson, S. (2010). Simplified Bayesian sensitivity analysis for mismeasured and unobserved confounders. Biometrics 66 1129–1137.
  • Herrnson, P. S. and Morris, I. L. (2007). Presidential campaigning in the 2002 congressional elections. Legis. Stud. Q. 32 629–648.
  • Holland, P. W. (1986). Statistics and causal inference. J. Amer. Statist. Assoc. 81 945–970.
  • Ichino, A., Mealli, F. and Nannicini, T. (2008). From temporary help jobs to permanent employment: What can we learn from matching estimators and their sensitivity? J. Appl. Econometrics 23 305–327.
  • Imai, K. and Yamamoto, T. (2008). Causal inference with measurement error: Nonparametric identification and sensitivity analyses of a field experiment on democratic deliberations. Paper presented at the 2008 Summer Political Methodology Meeting.
  • Imbens, G. W. (2003). Sensitivity to exogeneity assumptions in program evaluation. Am. Econ. Rev. Pap. Proc. 93 126–132.
  • Jacobson, G. C. (2003). The Politics of Congressional Elections, 6th ed. Longman, New York.
  • Jin, H. and Rubin, D. B. (2008). Principal stratification for causal inference with extended partial compliance. J. Amer. Statist. Assoc. 103 101–111.
  • Kadane, J. B. and Wolfson, L. J. (1998). Experiences in elicitation. Statistician 47 3–19.
  • Keele, L. J., Fogarty, B. and Stimson, J. A. (2004). The impact of presidential visits in the 2002 congressional elections. PS Polit. Sci. Polit. 34 971–986.
  • Manski, C. F. (1990). Nonparametric bounds on treatment effects. Am. Econ. Rev. Pap. Proc. 80 319–323.
  • Manski, C. F. (1993). Identification problems in the social sciences. Sociol. Method. 23 1–56.
  • Manski, C. F. (1995). Identification Problems in the Social Sciences. Harvard Univ. Press, Cambridge, MA.
  • Manski, C. F. (1997). Monotone treatment response. Econometrica 65 1311–1334.
  • Manski, C. F. (2003). Partial Identification of Probability Distributions. Springer, New York.
  • McCandless, L. C., Gustafson, P. and Levy, A. (2007). Bayesian sensitivity analysis for unmeasured confounding in observational studies. Stat. Med. 26 2331–2347.
  • Mealli, F. and Pacini, B. (2013). Using secondary outcomes to sharpen inference in randomized experiments with noncompliance. J. Amer. Statist. Assoc. 108 1120–1131.
  • Moon, H. R. and Schorfheide, F. (2012). Bayesian and frequentist inference in partially identified models. Econometrica 80 755–782.
  • Neustadt, R. E. (1960). Presidential Power. Wiley, New York.
  • Nutting, B. and Stern, A. H. (2001). CQ’s Politics in America: 2002 the 107th Congress. Congressional Quarterly Press, Washington.
  • Pearl, J. (1995). Causal diagrams for empirical research. Biometrika 82 669–710.
  • Pearl, J. (2000). Causality: Models, Reasoning, and Inference. Cambridge Univ. Press, Cambridge.
  • Richardson, T. S., Evans, R. J. and Robins, J. M. (2011). Transparent parametrizations of models for potential outcomes. In Bayesian Statistics 9 569–610. Oxford Univ. Press, Oxford.
  • Robins, J. (1986). A new approach to causal inference in mortality studies with a sustained exposure period—application to control of the healthy worker survivor effect. Math. Modelling 7 1393–1512.
  • Robins, J. M. (1989). The analysis of randomized and nonrandomized AIDS treatment trials using a new approach to causal inference in longitudinal studies. In Health Service Research Methodology: A Focus on AIDS (L. Sechreset, H. Freeman and A. Mulley, eds.) U.S. Public Health Service, Washington, DC.
  • Robins, J. M., Hernan, M. A. and Brumback, B. (2000). Marginal structural models and causal inference in epidemiology. Epidemiology 11 550–560.
  • Rosenbaum, P. R. (1987). Sensitivity analysis for certain permutation inferences in matched observational studies. Biometrika 74 13–26.
  • Rosenbaum, P. R. (2002). Observational Studies, 2nd ed. Springer, New York.
  • Rosenbaum, P. R. and Rubin, D. B. (1983). Assessing sensitivity to an unobserved covariate in an observational study with binary outcome. J. R. Stat. Soc., B 45 212–218.
  • Rubin, D. B. (1978). Bayesian inference for causal effects: The role of randomization. Ann. Statist. 6 34–58.
  • Rubin, D. B. (1990). Formal modes of statistical inference for causal effects. J. Statist. Plann. Inference 25 279–292.
  • Rubin, D. B. (2008). For objective causal inference, design trumps analysis. Ann. Appl. Stat. 2 808–804.
  • Sellers, P. J. and Denton, L. M. (2006). Presidential visits and midterm senate elections. Pres. Stud. Q. 36 410–432.
  • Western, B. and Jackman, S. (1994). Bayesian inference for comparative research. Am. Polit. Sci. Rev. 88 412–423.